Nice elliptic curves

In this work, we study elliptic curves of the form $E_L:y^2=x(x-1)(x-L)$, where $L^2-L+1\in {\left({\mathbb{Q}}^{\times }\right)}^2$ and $L\in \mathbb{Q}\setminus \{0,1\}$. We will show that for almost all quadratic extensions $K\mathrm{/}\mathbb{Q}$ except one we have $E_L(K)\cong \mathbb{Z}\mathrm{/}2\mathbb{Z}\oplus \mathbb{Z}\mathrm{/}2\mathbb{Z}$.